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Sums of sections, surface area measures, and the general Minkowski problem. (English) Zbl 1301.52009

Summary: For \(2\leq k\leq d -1\), the \(k\)-th mean section body, \(M_k(K)\), of a convex body \(K\) in \(\mathbb R^d\), is the Minkowski sum of all its sections by \(k\)-dimensional flats. We show that the characterization of these mean section bodies is equivalent to the solution of the general Minkowski problem, namely that of giving the characteristic properties of those measures on the unit sphere which arise as surface area measures (of arbitrary degree) of convex bodies. This equivalence arises from an analysis of Berg’s solution of the Christoffel problem. We see how the functions introduced by Berg yield an integral representation of the support function of \(M_k(K)\) in terms of the \((d+1 - k)\)-th surface area measure of \(K\). Our results are obtained using Fourier transform techniques which also yield a stability version of the fact that \(M_k(K)\) determines \(K\) uniquely.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A22 Random convex sets and integral geometry (aspects of convex geometry)
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Full Text: DOI Euclid